Question

a movie theater has a 27 - foot - high screen located 7 feet above your eye level. if you sit x feet back from the screen, your viewing angle, θ, is as given below.
θ = tan^(-1) (34/x) - tan^(-1) (7/x)
find the viewing angle, in radians, at distances of 5 feet, 10 feet, 15 feet, 20 feet, 25 feet.

Explanation:

Step1: Substitute x = 5 into the formula

$\theta=\tan^{- 1}\frac{34}{5}-\tan^{- 1}\frac{7}{5}$
$\theta=\tan^{- 1}(6.8)-\tan^{- 1}(1.4)$
Using a calculator, $\tan^{- 1}(6.8)\approx1.429$ radians and $\tan^{- 1}(1.4)\approx0.950$ radians.
$\theta\approx1.429 - 0.950=0.479$ radians

Step2: Substitute x = 10 into the formula

$\theta=\tan^{- 1}\frac{34}{10}-\tan^{- 1}\frac{7}{10}$
$\theta=\tan^{- 1}(3.4)-\tan^{- 1}(0.7)$
Using a calculator, $\tan^{- 1}(3.4)\approx1.287$ radians and $\tan^{- 1}(0.7)\approx0.611$ radians.
$\theta\approx1.287 - 0.611 = 0.676$ radians

Step3: Substitute x = 15 into the formula

$\theta=\tan^{- 1}\frac{34}{15}-\tan^{- 1}\frac{7}{15}$
$\theta=\tan^{- 1}(2.267)-\tan^{- 1}(0.467)$
Using a calculator, $\tan^{- 1}(2.267)\approx1.141$ radians and $\tan^{- 1}(0.467)\approx0.434$ radians.
$\theta\approx1.141-0.434 = 0.707$ radians

Step4: Substitute x = 20 into the formula

$\theta=\tan^{- 1}\frac{34}{20}-\tan^{- 1}\frac{7}{20}$
$\theta=\tan^{- 1}(1.7)-\tan^{- 1}(0.35)$
Using a calculator, $\tan^{- 1}(1.7)\approx1.040$ radians and $\tan^{- 1}(0.35)\approx0.337$ radians.
$\theta\approx1.040 - 0.337=0.703$ radians

Step5: Substitute x = 25 into the formula

$\theta=\tan^{- 1}\frac{34}{25}-\tan^{- 1}\frac{7}{25}$
$\theta=\tan^{- 1}(1.36)-\tan^{- 1}(0.28)$
Using a calculator, $\tan^{- 1}(1.36)\approx0.932$ radians and $\tan^{- 1}(0.28)\approx0.274$ radians.
$\theta\approx0.932-0.274 = 0.658$ radians

Answer:

When x = 5 feet, $\theta\approx0.479$ radians; when x = 10 feet, $\theta\approx0.676$ radians; when x = 15 feet, $\theta\approx0.707$ radians; when x = 20 feet, $\theta\approx0.703$ radians; when x = 25 feet, $\theta\approx0.658$ radians.